A series providing more explanation of how the atmosphere absorbs and emits radiation, including a simple model to provide insight. The model uses fictitious molecules pH2O and pCO2 (which have only a passing resemblance to the real molecules) to demonstrate some key points. Part Six even explains the real equations used.

Part Two – introducing a simple model, with molecules pH2O and pCO2 to demonstrate some basic effects in the atmosphere. This part – absorption only.

Part Three – the simple model extended to emission and absorption, showing what a difference an emitting atmosphere makes. Also very easy to see that the “IPCC logarithmic graph” is not at odds with the Beer-Lambert law.

Part Four – the effect of changing lapse rates (atmospheric temperature profile) and of overlapping the pH2O and pCO2 bands. Why surface radiation is not a mirror image of top of atmosphere radiation.

Part Five – a bit of a wrap up so far as well as an explanation of how the stratospheric temperature profile can affect “saturation”

Part Six – The Equations – the equations of radiative transfer including the plane parallel assumption and it’s nothing to do with blackbodies

Part Seven – changing the shape of the pCO2 band to see how it affects “saturation” – the wings of the band pick up the slack, in a manner of speaking

Part Eight – interesting actual absorption values of CO2 in the atmosphere from Grant Petty’s book

Part Nine – calculations of CO2 transmittance vs wavelength in the atmosphere using the 300,000 absorption lines from the HITRAN database

Part Ten – spectral measurements of radiation from the surface looking up, and from 20km up looking down, in a variety of locations, along with explanations of the characteristics

Part Twelve – The Curve of Growth – how absorptance increases as path length (or mass of molecules in the path) increases, and how much effect is from the “far wings” of the individual CO2 lines compared with the weaker CO2 lines

It seems clear to me and a very well designed question. I guess you should say infinite plates (to avoid questions about edge effects) and that there is no gravitational field.

Perhaps I can post an article here to ask for answers to your question?

I’m not sure of the answer myself without spending some time on it. First off, I thought that the temperature profile vs distance between the two plates (prior to CO2 being introduced) would be linear, now I am wondering. The heat transfer mechanism for the gas is conduction but as one side is hotter the density must increase, due to the ideal gas law, and this would change the conductivity.

The problem (as CO2 is introduced) is simplified vs the real atmosphere as the pressure is constant, which avoids the complexities of Doppler vs collisional broadening. Still, a great question is one where the problem is clear but you hesitate before trying to work out the answer.

I had a quibble with the cartoon at the top. The statement “There is no observational / empirical evidence for how much extra CO2 raises the global temperature” is counter to what I was taught in my university heat transfer course. My professor showed us how it could be done though the focus of the course was on metallurgical applications. At WUWT, I gave an example of a blast furnace where we calculate the increase in temperature of intervening gases to complete the energy balance. This is pretty clear empirical / observational evidence.

By the way, if you use the path length approximation and the emissivity curves in my metallurgical textbook as RTE approximations the plotted curve is approximately logarithmic. If you use the emissivity curves in my heat transfer text book it is more closely approximated by an hyperbola. The tail at high concentrations of CO2 is nearly parrallel to the concentration axis. That is at high concentrations of CO2 the effect of increased CO2 goes nearly to 0. As we have discussed in the past, this may not be relevant to the atmosphere as it is at constant pressure and the emissivity curves stop a 0C, hence in applying to the atmosphere one is going beyond the range of RTE’s used to generate the emissivity curves.
I wrote that post a while back. My wife packed all my school notes and put them into storage, so I don’t have quick access to them. I’ll see if I can find them and send you a link to the answer.

The metallurgical text is below. It may be out of print.
Schumann, Reinhardt, Metallurgical Engineering, Volume 1, Addison-Wesley,

You dismissed Frank’s valid criticism on the path length approximation on your site. That led you to make wrong conclusions on the effect of additional CO2. There’s another factor that adds to your error and that’s due to the fact that it’s enough for the warming influence of CO2 that the penetration changes in the upper part of the troposphere.

Then to your question. As you stated the details, there’s no energy transfer mechanism between the plates and the gas with 0% CO2. Thus in that case the heat transfer between the plates proceeds according to the Stephan-Boltzmann law but the gas may have any arbitrary temperature.

With a very small concentration of CO2 the gas gets essentially isothermal with the temperature ((300^4+350^4)/2)^0.25 = 327.8K because radiation from both plates affects equally the whole gap. (1% CO2 is already more than “very small”).

With very high concentration the temperature at each plate is close to the temperature of the plate and changes linearly in mass between these values. Linear in mass is not linear in distance due to the influence of temperature on the density. The heat transfer between the plates is reduced but not very much because the gas is transparent to most wavelengths.

With zero CO2 the gas and the plates interact via conduction as the only heat transfer mechanism. It will take a long time, but in the end there will be an approximately linear temperature profile with distance. I say approximately in case the temperature difference changes the conductivity of the gas.

I don’t disagree on that, but I took literally the total absence of all other energy transfer than radiative even on his point where it’s clearly unphysical.

Having the warmer plate at the top convection does not enter. Thus including only conduction is a valid alternative and that does lead slowly to a similar temperature profile as high concentration of CO2, but a low concentration of CO2 would still result in almost isothermal gas.

I have been idle this week for the first time in a long time, so have a chance ton answer your questio! My answer ( I continue to keep it hidden) is very VERY close to yours. The difference between what I have and what you have is entirely due to my method being an approximation. So. If anyone is looking for the exact answer, this is it. I still haven’t seen anyone with the approximate answer using path length approximations. Thanks again!

Short answer: Yes. If the light is absorbed, energy is transferred. If it’s reflected it isn’t. Light perpendicular to a clear water surface is mostly absorbed if the water is deep enough. Some short wavelength light is scattered by water more than longer wavelengths, which is what makes the ocean look blue. That’s also true for the sky.

All solar heating systems depend on the heating effect of the radiation. The efficiency of such systems has been calculated using spectral data. The results agree well with standard understanding. The share of visible light is a little less than 50% while all heating is due radiation with wavelengths less than 3µm.

These wavelengths have no physical link to electronic transition levels in pure water.
Any absorption of light is caused by particulates in the water for example sea water phytoplankton, seaweed and inorganic particulates.
The container of the water for example sea and pond bed may also absorb in the visible spectrum.

This curve as that you linked is for pure water without any impurities. A liquid can absorb at all wavelengths, it’s not restricted by the set of discrete excited states as a gas is when the pressure is not very high.

The link I gave is useful as it has a absolute vertical scale and it’s expanded enough to tell some interesting details like:

– blue light has a penetration depth of about 100 m, i.e. about a quarter of blue light penetrates more than 100 m in pure water
– the penetration depth of red light is only about 1 m
– near IR penetrates typically a fraction of one meter
– LW IR penetration depth is around 10 µm. That’s still enough to get some 1000 molecular layers deep into water.

All this is for pure water but the penetration depths are so long that the impurities really matter only for the short wavelength part of the visible spectrum.

The main conclusions are

– Most of the warming of solar radiation happens in the topmost meter or two.

– A significant fraction of warming happens even so at a depth of several meters and up to a few tens of meters. This fraction is significant because it causes warming at those depths and almost all that heat must escape through the surface. I.e. the net energy flux by convection and mixing is upwards and there must be times during which the layers at that depth are warmer than the layers above.

– Practically all interaction of radiation with water occurs deep enough to prevent direct transfer of energy from absorption of radiation to evaporation. I.e., the primary consequence of absorption is an addition to local thermal energy of water. That raises the water temperature and the rest follows as consequence of this increase in the temperature.

You’ve obviously never been scuba diving. The red end of the visible spectrum starts to go away almost immediately. The blue end can penetrate 100m or more. A flash is required even at moderate depths for photography in true colors.

Even the hyperphysics chart doesn’t show water in the visible band as perfectly transparent. It’s orders of magnitude more transparent than at longer and shorter wavelengths, but it’s not perfectly transparent anywhere. What you’re getting is absorption from the tails of the bands at longer and shorter wavelengths. The absorption cross section is small, but not zero.

That should be ‘ not necessarily absorption’ The general term is extinction. Extinction is due to scattering or absorption. Period. In this case, though, it’s definitely absorption.

There are strong scattering events particularly for the blue end.

If it were scattering instead of absorption, the blue end of the spectrum would penetrate less deeply than the red end. But of course it’s not. Rayleigh scattering is wavelength dependent. Scattering from particles much larger than the wavelength of the scattered light is Mie scattering and is largely wavelength independent. That’s why the sky is blue and clouds are white. The water droplets in clouds are too small to absorb enough of the red end of the visible spectrum to affect the color. But they absorb strongly in the IR.

Seriously, read Grant Petty’s book. It’s cheap if bought directly from the publisher. And read it, or read it again if you already have it, with the attitude that he knows more than you, because he does.

Not only is the sky blue, but the sun looks red when it’s low on the horizon. That’s mainly the Tyndall effect from scattering from particles on the same order of magnitude as the light wavelength. Wanna bet that the sun looks red looking up from about 10m down in clear water and gets redder as you go deeper?

The first (and dominating far) possibility is that it gets absorbed and that the energy goes immediately to heat the water.

The second possibility is that it’s absorbed by green plants and used to build organic material. The lifetime of organic material is mostly very short in oceans and most of that energy ends soon up as heat again. A very small part sinks to the bottom as organic material.

The third possibility is that the light penetrates all the way to bottom and warms first the bottom, but that heat ends almost totally in the water again.

The only case where the energy does not stay in the water is that where the light gets reflected or scattered back out of the ocean. The share of that is very small excluding that light that gets reflected already before it has really entered the ocean at all.

Your ability to selectively ignore anything that contradicts you and pull out something that might confirm your point is truly amazing.

Blue light is not absorbed by pure water.

This is standard physics as documented by the two links above.

No it isn’t. Apparently you are incapable of reading charts and graphs. The absorption coefficient of liquid water NEVER goes to zero either in your hyperphysics link or Pekka’s link. In Pekka’s link there is a table of absorption coefficients at six wavelengths in the visible. The transition at 401nm or 24940 cm-1 is aν1 + bν3; a+b=8. You can look up the rest yourself. Eyeballing the graph of the log of the absorption coefficient vs the log of the wavelength, it looks like the value at 400 nm is about 5E-5 cm-1 or a characteristic depth of 200m.

From Pekka’s link:

Water is almost perfectly transparent to ‘visible’ light,

[my emphasis]

Does the word ‘almost’ mean anything to you? Or do you just blank out any adjective that interferes with your ability to fool yourself.

Obviously, highly pure water does absorb in the near UV and Visible spectrum. It’s been measured by several different people. Well, probably not obviously to you as I have complete faith that you will be able to figure out a way to misinterpret this too and declare yourself right and everyone else wrong.

Neither DeWitt Payne nor I has argued against the importance of organic matter in attenuating UV, blue light or green light in natural water. We have only told that the the two links are on pure water and that even in pure water the attenuation is rather strong for all wavelengths. Your later link on the influence of the organic matter is about another issue.

The importance of the photosynthesis in the ocean energy balance can be estimated from the total primary production of oceans. The estimate for that is about 50 billion tons of carbon per year (see the presentation of Behrenfeld from the Ocean Productivity pages

The chemical energy content of biomass is roughly 40MJ/kg(carbon). From these numbers we can calculate the average power of biomass production as 63TW. The average solar energy that enters the oceans is approx. 90000TW. Thus photosynthesis takes only 0.07% of the solar energy that enters the oceans. This is far too little to justify your conclusions – and even less so because the lifetime of that organic material is short.

I knew you could do it! And as usual, you modify your position slightly while never admitting that your original contention was wrong and then move the goalposts. You do realize that only a small fraction of the solar energy absorbed by plant life for photosynthesis is converted to chemical energy. The rest ends up as thermal energy. And there really isn’t all that much plant life in the ocean on a concentration basis. If Wikipedia is to be believed, there are only 1-2 gigatons of carbon in the aquatic biosphere compared to 37,400 gigatons of inorganic carbon. Inorganic carbon is mostly in the form of bicarbonate ion which has a concentration in seawater of 145 mg/kg.

But let’s put that 10^8 difference into real perspective. The peak IR absorption depth is less than 1 μm. The minimum of the absorption coefficient curve in the dissertation represents a depth of ~220m at ~420 nm. That’s the same order of magnitude (about 100 m) that Pekka and I have been saying from the beginning and which you originally categorically denied. That’s indeed a difference of >2E08. But the ocean is a lot deeper than 220 m. In a shallow pond, some of the blue end of the spectrum will penetrate all the way to the bottom and be absorbed, unless the bottom of the pond is highly reflective. Even if it were reflected, more would be absorbed on the way back up to the surface. The end result being that nearly all the incoming solar radiation that isn’t initially reflected from the surface is absorbed and converted to thermal energy.

The atmosphere is almost transparent to short wave Solar radiation.

Now because this is a major part of the greenhouse theory its OK.
No effort is devoted to quantifying the ‘almost’ on this occasion.

Yet another attempt to deflect the conversation away from your error so you never have to admit you’re wrong about anything. And since irony must always increase, you’re wrong about this too. Total atmospheric absorption of SW radiation is prominently featured in the Trenberth et. al. energy balances. The clear sky spectrum of solar radiation at sea level is well known. What do you think the fuss about the ozone layer was all about?

The SURFRAD database includes measures of downwelling solar, upwelling (reflected) solar, direct normal solar and diffuse solar radiation in W/m² that run 24/7 at seven different sites in the continental US. But I’m sure you will manage to rationalize this as somehow not quantifying the almost.

There are areas in the ocean where the water is so pure that the absorption that you state as vanishingly small is the most important form of absorption. In most areas the impurities dominate in the absorption of blue light. In those areas the heating by the solar radiation is restricted to a thinner layer than in the pure water. Otherwise nothing changes significantly in the energy balance.

There’s nothing in this that would somehow contradict main stream understanding or IPCC reports. In particular the 0.07% that goes first to chemical energy makes no significant difference. (It’s important in other ways as discussed by climate scientists in many connections.)

Most people would regard this as a vanishingly small effect and effectively as near zero as dammit.

Most people would be wrong because the range of lengths involved are more than covered by the depth of the oceans. The UV absorption characteristic length at the peak near 55 nm is 10 nanometers . 100 meters is 10^10 greater than that. By the same logic, we should be able to ignore gravity because the strong force is more than 10^38 times greater and the electromagnetic force is more than 10^36 times greater.

Go back and read what I wrote earlier. It’s really easy to distinguish between mostly scattering and mostly absorption. If it’s scattering from particles the same size or smaller than the wavelength of light, the sun as viewed from below the surface will get redder as you go deeper because the blue end of the spectrum is scattered more efficiently. But it doesn’t. Red goes away first. I’ve been diving off the west coast of Catalina Island where the water is extremely clear and the reefs don’t start until about 100 feet down. Or at least it was that way nearly 50 years ago. It was very blue-green and got bluer as I went deeper. Flash photography at close range with wide angle or macro lenses is pretty much required for decent color even at fairly shallow depths. See the Sunlight section of the Wikipedia article on Underwater photography for example. Sure there’s some scattering, but it’s very small compared to absorption. Look at pictures of the Earth from space. The oceans appear to be a very dark blue.

[…] stumbled across Steve Carson’s wonderfully detailed and expository blog on atmospheric radiation and energy transfer effects this morning while googling something about Grant Petty’s textbook, A First Course in […]

Need to second DeWitt’s applause for Grant Petty’s book. I’m not “finished” with reading it, since that would mean doing all the problems, but I am working through it. But, if you want a quick intro to the same material, SOD’s blog right here is, from what I’ve found, the next best thing. Nothing quite replaces doing the problem sets, though.

As back-ground, I do believe that the earth is warming and that CO2 emissions exacerbate the natural cycles. However, I am puzzled by some of the widely spread explanations given, and especially the ‘earth energy budget’ diagrams widely distributed to explain current thinking. I think these are doing science a dis-service.

Can I first ask ‘the SoD panel’ a couple of questions:

1) According to these models how much energy is being emitted through the ‘atmospheric window’? What % is this of the input energy?
2) According to observed reality here (http://lasp.colorado.edu/~bagenal/3720/CLASS5/EarthBB.jpg) how much energy from the earths surface (at 12C~285K) do we see coming out? [FYI: the total output at 285K would be 374 w/m2. What % of this can you see coming out.]
3) How big is the atmospheric window in reality? Does it agree with the models.
4) How much radiation passes through an absorbing medium? [If you are stuck on this one, try holding a solid object, like a book up to the light … how much light comes through?]
5) If you have a candle, how closely can you put your hand to the side of it without burning? How closely can you put your hand to the top of it without burning?
6) If you have a hot engine, how hot does it feel if you put your hand under it? How hot does it feel if you put your hand on top (with the bonnet open)? How hot would it feel if you tried to unscrew the radiator cap (please do not try this)?
7) What are the dominant mechanisms for heat transfer in air in reality?
8) What happens to hot air? What happens when you heat air?
9) How much energy would the column of air above a 1m2 area of the earth’s surface store if it were stretched up by 5cm? [This might need a bit of science & googling.]
10) During the day, where do you think the energy from the surface that does not radiate out through the atmospheric window go?
11) During the night, what do you think happens?

Now you have had a chance to think, what are the flaws in those energy budgets?

a) The atmospheric window in reality is twice as big as shown in the models. See here.
b) In reality, no radiative heat transfer occurs between the clouds/atmosphere and the Earth’s surface. It is physically impossible.
c) Evaporation and convection are the dominant heat transfer mechanisms in reality in air. Not radiation.
d) When we heat air it expands and rises. When it contracts and falls it releases the stored energy. This is what keeps the Earth warm at night. Not radiation.

Th science appears to be fundamentally misrepresented (at least for these widely published energy budget diagrams).

In any energy budget you need to separately consider the flows occuring for Night and Day – without this we miss large energy flows into and out of the active daily energy reservoir. These major energy flows are completely missing. It is also probably worth separating the ‘atmosphere’ into the troposhpere (H2O, O3, CO2) layer, the tropopause (O3, CO2) layer and the ozone (03) layer. The interaction at these boundaries is quite fundamental to the overall picture.

Not sure what impact this has on the wider debate about the effect of CO2 on the climate, but if you cannot get the basic energy budget to correspond with reality you have to wonder.

I put in applause for Grant Petty’s book in January, but I have since discovered and studied a lot of Ray Pierrehumbert’s PRINCIPLES OF PLANETARY CLIMATE which, in addition to the nice Python code he offers, is better, because it offers a more comprehensive view — a thoroughly physical one — and subsumes much of the material in Petty’s work. Indeed, I own two copies of PRINCIPLES so I don’t need to carry a copy back and forth to my office.

This popular balance graphic and assorted variations are based on a power flux, W/m^2. A W is not energy, but energy over time, i.e. 3.4 Btu/eng h or 3.6 kJ/SI h. The 342 W/m^2 ISR is determined by spreading the average 1,368 W/m^2 solar irradiance/constant over the spherical ToA surface area. (1,368/4 =342) There is no consideration of the elliptical orbit (perihelion = 1,416 W/m^2 to aphelion = 1,323 W/m^2) or day or night or seasons or tropospheric thickness or energy diffusion due to oblique incidence, etc. This popular balance models the earth as a ball suspended in a hot fluid with heat/energy/power entering evenly over the entire ToA spherical surface. This is not even close to how the real earth energy balance works. Everybody uses it. Everybody should know better.

An example of a real heat balance based on Btu/h follows. Basically (Incoming Solar Radiation spread over the earth’s cross sectional area) = (U*A*dT et. al. leaving the lit side perpendicular to the spherical surface ToA) + (U*A*dT et. al. leaving the dark side perpendicular to spherical surface area ToA) The atmosphere is just a simple HVAC/heat flow/balance/insulation problem.

“Technically, there is no absolute dividing line between the Earth’s atmosphere and space, but for scientists studying the balance of incoming and outgoing energy on the Earth, it is conceptually useful to think of the altitude at about 100 kilometers above the Earth as the “top of the atmosphere.”
The top of the atmosphere is the bottom line of Earth’s energy budget, the Grand Central Station of radiation. It is the place where solar energy (mostly visible light) enters the Earth system and where both reflected light and invisible, thermal radiation from the Sun-warmed Earth exit. The balance between incoming and outgoing energy at the top of the atmosphere determines the Earth’s average temperature. The ability of greenhouses gases to change the balance by reducing how much thermal energy exits is what global warming is all about.”

ToA is 100 km or 62 miles. It is 68 miles between Denver and Colorado Springs. That’s not just thin, that’s ludicrous thin.

The GHE/GHG loop as shown on Trenberth Figure 10 is made up of three main components: upwelling of 396 W/m^2 which has two sub parts: 63 W/m^2 LWIR and 333 W/m^2 and downwelling of 333 W/m^2.

The 396 W/m^2 is calculated by inserting 16 C or 279K in the S-B BB equation, a calculation that does not actually exist in the real world. The result is 55 W/m^2 of power flux more than ISR entering ToA, an obvious violation of conservation of energy, i.e. created out of nothing. That should have been a warning.

ISR of 341 W/m^2 enter ToA, 102 W/m^2 are reflected by the albedo, leaving a net 239 W/m^2 entering ToA. 78 W/m^2 are absorbed by the atmosphere leaving 161 W/m^2 for the surface. To maintain the overall energy balance and a steady temperature (not really a requirement) 160 W/m^2 rises from the surface (0.9 residual in ground) as 17 W/m^2 convection, 80 W/m^2 latent and 63 W/m^2 LWIR (S-B BB 183 K, -90 C or emissivity = .16) = 160 W/m^2. All of the graphic’s power fluxes are now present and accounted for. The remaining and loop 333 W/m^2 are the spontaneous creation of an inappropriate application of the S-B BB equation violating conservation of energy.

But let’s press on.

The 333 W/m^2 upwelling/downwelling constitutes a 100% efficient perpetual energy loop violating thermodynamics. There is no net energy left at the surface to warm the earth and there is no net energy left in the troposphere to impact radiative balance at ToA.

The 333 W/m^2, 97% of ISR, upwells into the troposphere where it is allegedly absorbed/trapped/blocked by a miniscule 0.04% of the atmosphere. That’s a significant heat load for such a tiny share of atmospheric molecules and they should all be hotter than two dollar pistols.

Except they aren’t.

The troposphere is cold, -40 C at 30,000 ft, 9 km, < -60 C at ToA. Depending on how one models the troposphere, an evenly distributed average or weighted by layers from surface to ToA, the S-B BB equation for the tropospheric temperatures ranges from 150 to 250 W/m^2, a considerable, 45% to 75% of, less than 333.

(99% of the atmosphere is below 32 km where energy moves by convection/conduction/latent/radiation & where ideal S-B does not apply. Above 32 km the low molecular density does not allow for convection/conduction/latent and energy moves by S-B ideal radiation et. al.)

But wait!

The GHGs reradiate in all directions not just back to the surface. Say a statistical 33% makes it back to the surface that means 50 to 80 W/m^2. An even longer way away from the 333, 15% to 24% of.

But wait!

Because the troposphere is not ideal the S-B equation must consider emissivity. Nasif Nahle suggests CO2 emissivity could be around 0.1 or 5 to 8 W/m^2 re-radiated back to the surface. Light years from 333, 1.5% to 2.4% of.

But wait!

All of the above really doesn’t even matter since there is no net connection or influence between the 333 W/m^2 thermodynamically impossible loop and the radiative balance at 100 km ToA. Just erase this loop from the graphic and nothing else about the balance changes.

BTW 7 of the 8 reanalyzed (i.e. water board the data until it gives up the “right” answer) data sets/models show more power flux leaving OLR than entering ASR ToA or atmospheric cooling. Obviously those seven data sets/models have it completely wrong because there can’t possibly be any flaw in the GHE theory.

The GHE greenhouse analogy/theory not only doesn’t apply to the atmosphere, it doesn’t even apply to warming a real greenhouse. (“The Discovery of Global Warming” Spencer Weart) In a real greenhouse the physical barrier of walls, glass, plastic trap the convective heat, not some kind of handwavium glassy, transparent, radiative thermal diode.

The surface of the earth is warm for the same reason a heated house is warm in the winter: Q = U * A * dT, the energy flow/heat resisting blanket of the insulated walls. Same for the atmospheric blanket. A blanket works by Q = U * A * dT, not S-B BB. The composite thermal conductivity of that paper-thin atmosphere, conduction, convection, latent, LWIR, resists the flow of energy, i.e. heat, from surface to ToA and to make that energy flow (heat) requires a temperature differential, 213 K ToA and 288 K surface = 75 C. The atmosphere is just a basic HVAC system boundary analysis.

Open for rebuttal. If you can explain how this upwelling/downwelling/”back” radiation actually really works be certain to copy Jennifer Marohasy as she has posted a challenge for such an explanation.

As you have posted a comment in another article, which has brevity and clarity of mistakes relating to the equations of heat transfer on its side, I will await your reply there before venturing to respond to your essay here.